Dynamic Asset Allocationwith Event Risk

THE JOURNAL OF FINANCE VOL. LVIII, NO.1 FEB. 2003 Dynamic Asset Allocationwith Event Risk JUN LIU, FRANCISA. LONGSTAFF and JUN PANn ABSTRACT Major events often trigger abrupt changes in stock prices and volatility.We study the implications of jumps in prices and volatility on investment strate-gies. Using the event-risk framework of Du⁄e, Pan, and Singleton (2000), we provide analytical solutions to the optimal portfolio problem. Event risk dra-matically a¡ects the optimal strategy. An investor facing event risk is less willing to take leveraged or short positions.The investor acts as if some por-tion of his wealth may become illiquid and the optimal strategy blends both dynamicandbuy-and-hold strategies. Jumpsinpricesandvolatilitybothhave important e¡ects. ONEOFTHEINHERENTHAZARDS ofinvestingin¢nancialmarketsistheriskofamajor eventprecipitatingasuddenlargeshocktosecurity pricesandvolatilities.There are manyexamples of this type ofevent, including, most recently, the September 11,2001, terrorist attacks. Other recent examples includethe stock marketcrash ofOctober19,1987,inwhichtheDowindexfellby508points,theOctober27,1997, drop in the Dow index by more than 554 points, and the £ight to quality in the aftermath of the Russian debt default where swap spreads increased on August 27, 1998, by more than 20 times their daily standard deviation, leading to the downfall of Long Term Capital Management and many other highly leveraged hedgefunds.Eachoftheseeventswasaccompaniedbymajorincreasesinmarket volatility.1 The risk of event-related jumps in security prices and volatility changes the standard dynamic portfolio choice problem in several important ways. In the standard problem, security prices are continuous and instantaneous returns have in¢nitesimal standard deviations; an investor considers only small local changesinsecuritypricesinselectingaportfolio.Withevent-relatedjumps,how-ever, the investor must also consider the e¡ects of large security price andvola- n Liuand Longsta¡ are with the Anderson School at UCLA and Pan is with the MIT Sloan School of Management.We are particularly grateful for helpful discussions withTony Bernar-do and Pedro Santa-Clara, for the comments of Jerome Detemple, Harrison Hong, Paul P£ei-derer, Raman Uppal, and participants at the 2001 Western Finance Association meetings, and for the many insightful comments and suggestions of the editor Richard Green and the referee. All errors are our responsibility. 1For example, the VIX index of S&P 500 stock index option implied volatilities increased 313 percent on October 19, 1987, 53 percent on October 27, 1997, and 28 percent on August 27, 1998. 231 232 TheJournalofFinance tility changes when selecting a dynamic portfolio strategy. Since the portfolio that is optimal for large returns need not be the same as that for small returns, this creates a strong con£ict that must be resolved by the investor in selectinga portfolio strategy. This paper studies the implications of event-related jumps in security prices andvolatilityonoptimal dynamicportfolio strategies. In modelingevent-related jumps, we use the double-jump framework of Du⁄e, Pan, and Singleton (2000). This framework is motivated by evidence by Bates (2000) and others of the exis-tence of volatility jumps, and has received strong empirical support from the data.2 Inthismodel,boththesecuritypriceandthevolatilityofitsreturnsfollow jump-di¡usion processes. Jumps are triggered by a Poisson event which has an intensity proportional to the level of volatility.This intuitive framework closely parallels thebehaviorof actual ¢nancial markets and allowsus to studydirectly the e¡ects ofevent riskon portfolio choice. To make the intuition behind the results as clear as possible, we focus on the simplest case where an investor with power utility over end-of-period wealth al-locates his portfolio between a riskless asset and a risky asset that follows the double-jump process. Because of thetractability providedby the a⁄ne structure of the model, we are able to reduce the Hamilton^Jacobi^Bellman partial di¡er-ential equation for the indirect utility function to a set of ordinary di¡erential equations.This allows us to obtain an analytical solution for the optimal port-folio weight. In the general case, the optimal portfolio weight is given by solving a simple pair of nonlinear equations. In a number of special cases, however, closed-form solutions for the optimal portfolioweight are readilyobtained. The optimal portfolio strategy inthepresence ofevent risk has many interest-ing features. One immediate e¡ect of introducing jumps into the portfolio pro-blem is that return distributions may display more skewness and kurtosis. While this has an important in£uence on the portfolio chosen, the full implica-tions of event risk for dynamic asset allocation run much deeper.We show that the threat of event-related jumps makes an investor behave as if he faced short-sellingandborrowingconstraintseventhoughnoneareimposed.Thisresultpar-allels Longsta¡ (2001) where investors facing illiquid or nonmarketable assets restrict their portfolio leverage. Interestingly, we ¢nd that the optimal portfolio is ablendof the optimalportfolioforacontinuous-timeproblem andthe optimal portfolio for a static buy-and-hold problem. Intuitively, this is because when an event-relatedjumpoccurs,theportfolioreturnisonthesameorderofmagnitude as the return that would be obtained from abuy-and-hold portfolio over some ¢-nite horizon. Since these two returns have the same e¡ect on terminal wealth, their implications for portfolio choice are indistinguishable, and event risk can beinterpretedorviewedasaformofliquidityrisk.Thisperspectiveprovidesnew insights into the e¡ects ofevent riskon ¢nancial markets. To illustrate our results, we provide two examples. In the ¢rst, we consider a model where the riskyasset follows a jump-di¡usion process with deterministic 2 For example, see the extensive recent study by Eraker, Johannes, and Polson (2000) of the double-jump model. DynamicAssetAllocationwithEventRisk 233 jump sizes, but where return volatility is constant. This special case parallels Merton (1971), who solves for the optimal portfolio weight whenthe riskless rate follows a jump-di¡usion process. We ¢nd that an investor facing jumps may chooseaportfolioverydi¡erentfromtheportfoliothatwouldbeoptimalifjumps did not occur. In general, the investor holds less of the risky asset when event-related price jumps can occur.This is true even when only upward price jumps canoccur.Intuitively,thisisbecausethee¡ectofjumpsonreturnvolatilitydom-inates the e¡ect of the resulting positive skewness. Because event risk is con-stant over time in this example, the optimal portfolio does not depend on the investor’s horizon. In the second example, we consider a model where both the riskyasset and its return volatility follow jump-di¡usion processes with deterministic jump sizes. The stochastic volatility model studied by Liu (1999) can be viewed as a special caseof this model.AsinLiu,theoptimalportfolioweightdoesnotdependonthe level of volatility.The optimal portfolio weight, however, does depend on the in-vestor’s horizon, since the probabilityof an event is time varying through its de-pendence on the level of volatility. We ¢nd that volatility jumps can have a signi¢cant e¡ect on the optimal portfolio above and beyond the e¡ect of price jumps. Surprisingly, investors may even choose to hold more of the risky asset when there are volatility jumps than otherwise. Intuitively, this means that the investor can partiallyhedge the e¡ects of volatility jumps on his indirect utility through the o¡setting e¡ects of price jumps. Note that this hedging behavior arises because of the static buy-and-hold component of the investor’s portfolio problem; this staticjump-hedgingbehaviordi¡ersfundamentally fromtheusual dynamic hedging of state variables that occurs in the standard pure-di¡usion portfolio choice problem. We provide an application of the model by calibrating it to historical U.S. data and examining its implications for optimal portfolio weights. The results show that even when large jumps are very infrequent, an investor still ¢nds it optimal to reduce his exposure to the stock market signi¢cantly.These results suggestapossiblereasonwhyhistoricallevelsofstockmarketparticipationhave tendedtobelowerthanwouldbe optimalin manyclassicalportfoliochoicemod-els. While volatility jumps are qualitatively important for optimal portfolio choice, the calibrated exercise shows that they generally have less impact than pricejumps. Sincethe originalworkby Merton(1971),theproblemof portfoliochoiceinthe presence of richer stochastic environments has become a topic of increasing in-terest.RecentexamplesofthisliteratureincludeBrennan,Schwartz,andLagna-do (1997) on asset allocation with stochastic interest rates and predictability in stock returns, Kim and Omberg (1996), Campbell and V|ceira (1999), Barberis (2000), and Xia (2001) on predictability in stock returns (with or without learn-ing), Lynch (2001) on portfolio choice and equity characteristics, Schroder and Skiadas (1999) on a class of a⁄ne di¡usion models with stochastic di¡erential utility, Balduzzi and Lynch (1999) ontransactioncosts and stock return predict-ability, and Brennan and Xia (1998), Liu(1999), Wachter (1999), Campbell and V|ceira (2001) on stochastic interest rates, and Ang and Bekaert (2000) on 234 TheJournalofFinance time-varying correlations. Aase (1986), and Aase and Òksendal (1988) study the properties of admissible portfolio strategies in jump di¡usion contexts. Aase (1984), Jeanblanc-Picque and Pontier (1990), and Bardhan and Chao (1995) provide more general analyses of portfolio choice when asset price dynamics are discontinuous. Although Merton (1971), Common (2000), and Das and Uppal (2001) study the e¡ects of price jumps and Liu (1999), Chacko andV|ceira (2000), and Longsta¡ (2001) study the e¡ects of stochastic volatility, this paper contributestotheliteraturebybeingthe¢rsttostudythee¡ectsofevent-related jumps inboth stock prices andvolatility.3 The remainder of this paper is organized as follows. Section I presents the event-riskmodel.SectionIIprovidesanalyticalsolutionstotheoptimalportfolio allocationproblem.SectionIIIpresentstheexamplesandprovidesnumerical re-sults. Section IVcalibrates the model and examines the implications foroptimal portfolio choice. Section V summarizes the results and makes concluding re-marks. I. The Event-Risk Model Weassumethattherearetwoassetsintheeconomy.The¢rstisarisklessasset paying a constant rate of interest r.The second is a riskyasset whose price St is subjecttoevent-relatedjumps.Speci¢cally,thepriceoftheriskyassetfollowsthe process pffiffiffiffiffiffi dSt ¼ ðrþ ZVt mlVtÞStdt þ VtStdZ1t þXtSt dNt; ð1Þ pffiffiffiffiffiffi dVt ¼ ða bVt klVtÞdt þ s VtdZ2t þ YtdNt ð2Þ where Z1 and Z2 are standard Brownian motions with correlation r,V is the in-stantaneous variance of di¡usive returns, and N is a Poisson process with sto-chastic arrival intensity lV.The parameters a, b, k, l, and s are all assumed to be nonnegative.The variable X is a random price-jump size with mean m, and is assumed to have support on (1, N) which guarantees the positivity (limited liability) of S. Similarly,Y is a random volatility-jump size with mean k, and is assumed to have support on [0, N) to guarantee thatVremains positive. In gen-eral, the jump sizes X andYcan be jointly distributed with nonzero correlation. The jump sizes X andY are also assumed to be independent across jump times and independent of Z1, Z2, and N. Giventhesedynamics,thepriceoftheriskyassetfollowsastochastic-volatility jump-di¡usionprocessandisdrivenbythreesourcesofuncertainty:(1)di¡usive priceshocksfromZ1,(2)di¡usivevolatilityshocksfromZ2,and(3)realizationsof the Poisson process N. Since a realization of N triggers jumps in both S andV, a realizationofNhas thenaturalinterpretationofa ¢nancialeventa¡ectingboth prices and market volatilities. In this sense, this model is ideal for studying the 3 Wu(2000) studies the portfolio choice problem in a model where there are jumps in stock prices but not volatility, but does not provide a veri¢able analytical solution for the optimal portfolio strategy. DynamicAssetAllocationwithEventRisk 235 e¡ects of event risk on portfolio choice. Because the jump sizes X andYare ran-dom, however, itispossible for the arrivalof anevent toresult in a largejumpin S and only a small jump inV, or a small jump in S and a large jump inV.This feature is consistent with observed market behavior; although ¢nancial market events are generally associated with large movements in both prices and volati-lity, jumps in only prices or only volatility can occur. Since m is the mean of the price-jump size X, the term mlVS in equation (1) compensates for the instanta-neousexpected returnintroducedby thejump component of thepricedynamics. As a result, the instantaneous expected rate of return equals the riskless rate r plusariskpremiumZV. ThisformoftheriskpremiumfollowsfromMerton(1980) and is also used by Liu (1999), Pan (2002), and many others. Note that the risk premium compensates the investor for both the risk of di¡usive shocks and the riskof jumps.4 These dynamics also imply that the instantaneous varianceV follows a mean-reverting square-root jump-di¡usion process.The Heston (1993) stochastic-vola-tility modelcan be obtained as a specialcase of this model by imposing the con-dition that l50, which implies that jumps do not occur. Liu (1999) provides closed-form solutions to the portfolio problem for this special case.5 Also nested as special cases are the stochastic-volatility jump-di¡usion models of Bates (2000) and Bakshi, Cao, and Chen (1997). Again, since k is the meanof the volati-lity jump sizeY, klV in the drift of the process forVcompensates for the jump component involatility. Thisbivariatejump-di¡usion model is an extendedversionof the double-jump model introduced by Du⁄e et al. (2000). Note that this model falls within the af-¢ne classbecause of thelinearityof the drift vector, di¡usion matrix, and inten-sity process in the state variableV. The double-jump framework has received a signi¢cant amount of empirical support because of the tendency for both stock pricesandvolatilitytoexhibitjumps.Forexample,arecentpaperbyErakeretal. (2000)¢ndsstrongevidenceofjumpsinvolatilityevenafteraccountingforjumps in stock returns.6 Du⁄e et al. also show that the double-jump model implies vo-latility ‘‘smiles’’or skews for stockoptions that closely matchthevolatilityskews observed inoptions markets. II. Optimal Dynamic Asset Allocation In this section, we focus on the asset allocation problem of an investor with power utility 4 Although the risk premium could be separated into the two types of risk premia, the port-folio allocation between the riskless asset and the risky asset in our model is independent of this breakdown. If options were introduced into the market as a second risky asset, however, this would no longer be true (see Pan (2002)). 5 See Chacko and V|ceira (2000) and Longsta¡ (2001) for solutions to the dynamic portfolio problem for alternative stochastic volatility models. 6 Similar evidence is also presented in Bates (2000), Pan (2002), and others. 7 See also Bakshi et al. (1997) and Bates (2000) for empirical evidence about the importance of jumps in option pricing. ... - tailieumienphi.vn